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MISCELLANEOUS TABLES.

TIME.

208. Time is used in measuring periods of duration; as years, days, minutes, etc.

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1. The period of time required for the earth to revolve about the sun, or the true year, is 365 da. 5 h. 48 min. 49.7 sec., or nearly 365 days. This exceeds the common year by 5 h. 48 min. 49.7 sec. (not quite a quarter of a day). Instead of reckoning this day each year, we consider the year as 365 days, and add the, or 1 day, at the end of every fourth, bissextile, or leap year, to the month of February. But since the excess of the true year over the common year is not quite a quarter of a day, and the difference would amount to about a day in 400 years, we discard one leap year in every 400 years; that is, centennial years exactly divisible by 400, and other years exactly divisible by 4, are considered as leap years. Centennial years divisible by 4, but not by 400, are not leap years. Thus, the year 1900 will not be a leap year, but the year 1600 was.

2. Julius Cæsar, the Roman Emperor, decreed that the year should consist of 365 days 6 hours; that the 6 hours should be disregarded for 3 successive years, and an entire day be added to every fourth year. This day was inserted in the calendar between the 24th and 25th days of February, and is called the intercalary day. As the Romans counted the days backward from the first day of the following month, the 24th of February was called by them sexto calendas Martii, the sixth before the calends of March. The intercalary day which followed this was called bis-sexto calendas Martii; hence the name bissextile.

3. In 1582 the error in the calendar as established by Julius Cæsar had increased to 10 days; that is, too much time had been reckoned as a year, until the civil year was 10 days behind the solar year. To correct this error, Pope Gregory decreed that 10 entire days should be stricken from the calendar, and that the day following the 3d day of October, 1582, should be the 14th. This brought the vernal equinox at March 21 - the date on which it occurred in the year 325, at the time of the Council of Nice.

4. The year as established by Julius Cæsar is sometimes called the Julian Year; and the period of time in which it was in force, namely, from 46 years B.C. to 1582, is called the Julian Period.

5. The year as established by Pope Gregory is called the Gregorian Year, and the calendar now used is the Gregorian Calendar.

6. Most Catholic countries adopted the Gregorian Calendar soon after it was established. Great Britain, however, continued to use the Julian Calendar until 1752. At this time the civil year was 11 days behind the solar year. To correct this error, the British Government decreed that 11 days should be stricken from the calendar, and that the day following the 2d day of September, 1752, should be the 14th.

7. Time before the adoption of the Gregorian Calendar is called Old Style (O. S.), and since, New Style (N. S.). In Old Style the year commenced March 25, and in New Style it commences January 1.

8. Russia still reckons time by Old Style, or the Julian Calendar; hence her dates are now 12 days behind ours.

9. The centuries are numbered from the commencement of the Christian era; the months from the commencement of the year; the days from the commencement of the month, and the hours from the commencement of the day (12 o'clock, midnight). Thus, May 23, 1893, 9 o'clock A.M., is the 9th hour of the 23d day of the 5th month of the 93d year of the 19th century.

10. In most business transactions 30 days are called 1 month. For many purposes 4 weeks constitute a month.

11. The number of days in each calendar month may be easily remembered by committing the following lines to memory:

"Thirty days hath September,

April, June, and November;

All the rest have thirty-one,

Save February, which alone

Hath twenty-eight; and one day more

We add to it one year in four."

EXAMPLES.

1. Reduce 365 da. 5 h. 48 min. 46 sec. to seconds.

2. Reduce 31556926 seconds to days.

3. In 3114061 seconds how many weeks are there?

4. In 5 wk. 1 da. 1 h. 1 min. 1 sec., how many seconds are there?

5. How many times does a clock pendulum, beating seconds, vibrate in one day? Ans. 86400. 6. If a man takes 1 step a yard long in a second, in how long a time will he walk 10 miles?

Ans. 4 h. 53 min. 20 sec.

7. In a lunar month of 29 da. 12 h. 44 min. 3 sec., how many seconds are there? Ans. 2551443. 8. How much time will a person gain in 40 years, by rising 45 minutes earlier every day?

Ans. 456 da. 13 h. 30 min

CIRCULAR OR ANGULAR MEASURE.

209. Circular or Angular Measure, or Circular motion, is used principally in surveying, navigation, astronomy, and geography, to measure arcs of angles or circles, for reckoning latitude and longitude, determining locations of places and vessels, and computing difference of time.

A Circle is a plane figure bounded by a curved line called the circumference, every point of Circumference mference which is equally distant from a point

Diameter

Radius

Circle

within called the center.

That part of the circumference which is included within the lines which form the angle, is the measure of the angle. The Radius of a circle is a line extending from its center to any point in the circumference. It is one half the diameter.

The Diameter of a circle is a line passing through its center, and terminated at both ends by the cir

cumference. The circumference of every circle is 3.1416+ times the diameter.

following rules:

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Hence we have the

RULE. To find the circumference of a circle, multiply the diameter by 3.1416.

To find the diameter of a circle, multiply the circumference by .3183.

The rules for finding the area of a circle are derived from principles of geometry which cannot be here explained.

I. To find the area of a circle when both diameter and circumference are given, multiply the diameter by the circumference, and divide the product by 4.

II. When the diameter is given, multiply the square of the diameter by .7854.

III. When the circumference is given, multiply the square of the circumference by .07958.

The square of a number is the product of that number multiplied by itself.

Every circle, great or small, is divisible into the same number of equal parts, as quarters, called quadrants, twelfths, called signs, 360ths, called degrees, etc. Consequently the parts of different circles, although having the same names, are of different lengths. The unit is the degree.

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1. Minutes of the earth's circumference are called geographic or nautical miles. The length of a degree of longitude on the earth's surface at the equator is 69.16 miles.

2. The denomination, sign (30°), is confined exclusively to astronomy.

3. Degrees are not strictly divisions of a circle, but of the space about a point in any plane.

4. 90° make a quadrant, or right angle, 60° a sextant, or of a circle.

EXAMPLES.

1. Reduce 10 S. 10° 10' 10" to seconds.

2. Reduce 1116610" to signs.

3. How many degrees are there in 11400 geographic or nautical miles? Ans. 190°. 4. If one degree of the earth's circumference is 691 statute miles, how many statute miles are there in 11400 geographic miles, or 190 degrees? Ans. 13148. 5. How many minutes, or nautical miles, are there in the circumference of the earth? Ans. 21600' or mi. 6. A ship during 4 days' storm at sea changed her longitude 397 geographic miles. How many degrees and minutes did she change? Ans. 6° 37'.

7. The diameter of a circle is 113, and the circumference 355. What is the area?

Ans. 10028.75.

8. What is the area of a circular island whose diameter is 1 mile?

9. A man has a circular garden requiring 84 rods of fencing to inclose it. How much land is there in the Ans. 3 A. 81.5+ P.

garden?

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The terms folio, quarto, octavo, duodecimo, etc., indicate the number of leaves into which a sheet of paper is folded. When a sheet is folded into 2, 4, 8, 12, 16, 18, or 24 forms, it is called, respectively, a folio, quarto (4to), octavo (8vo), duodecimo (12mo), 16mo, 18mo, or 24mo.

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